Fe 0.02 O 3 by the Mean Field Theory
7. Giaever tunnelling
Figure 21 depicts the schematic of a superconducting quantum interference device (SQUID) where two superconductors S1and S2are separated by thin insula- tors. A small flux through the SQUID creates a phase difference in the two super- conductors (see discussion onΔk above) leading to the flow of supercurrent. If an initial phase,δ0exists between the superconductors. Then, this phase difference after application of flux is from Eq. (42),δa¼δ0þeΦℏ0across top insulator and δa¼δ0�eΦℏ0 across bottom insulator (see Figure 21). This leads to currents Ja¼I0sinδaand Jb ¼I0sinδbthrough top and down insulators. The total current J¼JaþJb ¼2I0sinδ0coseΦℏ0. This accumulates charge on one side of SQUID and leads to a potential difference between the two superconductors. Therefore, the flux is converted to a voltage difference. The voltage oscillates as the phase differenceeΦℏ0 goes in integral multiples ofπfor every flux quantaΦ0. SQUID is the most sensitive magnetic flux sensor currently known. The SQUID can be seen as a flux to voltage converter, and it can generally be used to sense any quantity that can be transduced into a magnetic flux, such as electrical current, voltage, position, etc. The extreme sensitivity of the SQUID is utilized in many different fields of applications, includ- ing biomagnetism, materials science, metrology, astronomy and geophysics.
Figure 21 depicts the schematic of a superconducting quantum interference device (SQUID) where two superconductors S1and S2are separated by thin insula- tors. A small flux through the SQUID creates a phase difference in the two super- conductors (see discussion onΔk above) leading to the flow of supercurrent. If an initial phase,δ0exists between the superconductors. Then, this phase difference after application of flux is from Eq. (42),δa¼δ0þeΦℏ0across top insulator and δa ¼δ0�eΦℏ0across bottom insulator (see Figure 21). This leads to currents Ja¼I0sinδaand Jb¼I0sinδbthrough top and down insulators. The total current J¼JaþJb¼2I0sinδ0coseΦℏ0. This accumulates charge on one side of SQUID and leads to a potential difference between the two superconductors. Therefore, the flux is converted to a voltage difference. The voltage oscillates as the phase differenceeΦℏ0 goes in integral multiples ofπfor every flux quantaΦ0. SQUID is the most sensitive magnetic flux sensor currently known. The SQUID can be seen as a flux to voltage converter, and it can generally be used to sense any quantity that can be transduced into a magnetic flux, such as electrical current, voltage, position, etc. The extreme sensitivity of the SQUID is utilized in many different fields of applications, includ- ing biomagnetism, materials science, metrology, astronomy and geophysics.
6. Meissner effect
When a superconductor placed in an magnetic field is cooled below its critical Tc, we find it expel all magnetic field from its inside. It does not like magnetic field in its interior. This is shown in Figure 22.
German physicists Walther Meissner and Robert Ochsenfeld discovered this phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin and lead samples. The samples, in the presence of an applied magnetic field, were cooled below their superconducting transition temperature, whereupon the samples canceled nearly all interior magnetic fields. A supercon- ductor with little or no magnetic field within it is said to be in the Meissner state.
The Meissner state breaks down when the applied magnetic field is too large.
Superconductors can be divided into two classes according to how this breakdown occurs. In type-I superconductor if the magnetic field is above certain threshold Hc, no expulsion takes place. In type-II superconductors, raising the applied field past a critical value Hc1leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no
Figure 22.
Depiction of the Meissner effect whereby the magnetic field inside a superconductor is expulsed when we cool it below its superconducting temperatureTc.
resistance to the electric current as long as the current is not too large. At a second critical field strength Hc2, no magnetic field expulsion takes place. How can we explain Meissner effect?
We talked about how wave packets shuttle back and forth in local potentials and get bound by phonons to form a BCS molecule. In the presence of a magnetic field, they do not shuttle. Instead, they do cyclotron motion with frequencyω¼eBm. This is shown in Figure 23. At a field of about 1 Tesla, this is about 1011rad/s. Recall our packets had a width ofωD�1013Hz and the shuttling time of packets was 10�13s, so that the offsets in a packet do not evolve much in the time the packet is back. But, when we are doing cyclotron motion, it takes 10�11s (at 1 T field) to come back, and by that time, the offsets evolve a lot, which means poor binding. It means in the presence of magnetic field we cannot bind well. Therefore, physics wants to get rid of magnetic field, bind and lower the energy. Magnetic field hurts binding and therefore it is expelled. But if the magnetic field is increased, then the cyclotron frequency increases, and at a critical value, our packet returns home much faster, allowing for little offset evolution; therefore, we can bind and there is no need to expulse the magnetic field. This explains the critical field.
7. Giaever tunnelling
When we bring two metals in proximity, separated by a thin-insulating barrier, apply a tiny voltage nd then the current will flow in the circuit. There is a thin- insulating barrier, but electrons will tunnel through the barrier. Now, what will happen if these metals are replaced by a superconductor? These are a set of exper- iments carried out by Norwegian-American physicist Ivar Giaever who shared the Nobel Prize in Physics in 1973 with Leo Esaki and Brian Josephson “for their discoveries regarding tunnelling phenomena in solids”. What he found was that if one of the metals is a superconductor, the electron cannot just come in, as there is an energy barrier ofΔ, the superconducting gap. Your applied voltage has to be at least as big asΔfor tunneling to happen. This is depicted in Figure 24. Let us see why this is the case.
Recall in our discussion of superconducting state that we had 2p pockets of which p were empty. We had a binding energy of�4ℏdω2dNp2. What will it cost to
Figure 23.
(A) Depiction on how packets shuttle back and forth in local potential and (B) how they execute a cyclotron motion in the presence of a magnetic field.
bring in an extra electron as shown in Figure 25? It will go in one of the empty pockets, and then we have only p�1 pockets left to scatter to reducing the binding energy to�4ℏd2Np p�1ωdð Þwith a changeΔ¼4ℏdω2dNp. Therefore, the new electron raises the energy byΔ, and therefore to offset this increase of energy, we have to apply a voltage as big asΔfor tunneling to happen.
8. HighTcin cuprates
High-temperature superconductors (abbreviated high-Tcor HTS) are materials that behave as superconductors at unusually high temperatures. The first high-Tc
superconductor was discovered in 1986 by IBM researchers Georg Bednorz and K.
Alex Müller, who were awarded the 1987 Nobel Prize in Physics “for their impor- tant break-through in the discovery of superconductivity in ceramic materials”.
Whereas “ordinary” or metallic superconductors usually have transition temperatures (temperatures below which they are superconductive) below 30 K (243.2°C) and must be cooled using liquid helium in order to achieve
superconductivity, HTS have been observed with transition temperatures as high as 138 K (135°C) and can be cooled to superconductivity using liquid nitrogen.
Compounds of copper and oxygen (so-called cuprates) are known to have HTS properties, and the term high-temperature superconductor was used interchange- ably with cuprate superconductor. Examples are compounds such as lanthanum strontium copper oxide (LASCO) and neodymium cerium copper oxide (NSCO).
Figure 24.
(A) Depiction on how a tiny voltage between two metals separated by an insulating barrier generates current that goes through an insulating barrier through tunneling. (B) If one of the metals is a superconductor, then the applied voltage has to be at least as big as the superconducting gap.
Figure 25.
Depiction on how upon tunneling an extra electron, shown in dashed lines, enters the superconducting state.
Let us take lanthanum copper oxide La2CuO4, where lanthanum donates three electrons and copper donates two and oxygen accepts two electrons and all valence is satisfied. The Cu is in state Cu2+with electrons in d9configuration [11]. d orbitals are all degenerated, but due to crystal field splitting, this degeneracy is broken, and dx2y2 orbital has the highest energy and gets only one electron (the ninth one) and forms a band of its own. This band is narrow, and hence all k states get filled by only one electron each and form Wannier packets that localize electrons on their respective sites. This way electron repulsion is minimized, and in the limit U≫t (repulsion term in much larger than hopping), we have Mott insulator and antiferromagnetic phase.
When we hole/electron dope, we remove/add electron to dx2y2band. For example, La2xSrxCuO4is hole doped as Sr has valence 2 and its presence further removes electrons from copper. Nd2xCexCuO4is electron doped as Ce has valence 4 and its presence adds electrons from copper. These extra holes/electrons form packets. When we discussed superconductivity, we discussed packets of widthωd. In d-bands, a packet of this width has many more k states as d-bands are narrow and only 1–2 eV thick. This means N (k-points in a packet) is very large and we have much larger gapΔand Tc. This is a way to understand high Tc, d-wave packets with huge bandwidths. This is as shown in Figure 26. As we increase doping and add
Figure 26.
(A) Depiction on how in narrowdband the electron wave packet comprises allkpoints to minimize repulsion.
(B) The wave packet in two dimensions with wave packet formed from superposition of k-points (along the k- direction) inside the Fermi sphere.
Figure 27.
The characteristic phase diagram of a highTccuprate. Shown are the antiferromagnetic insulator (AF) phase on the left anddome-shaped superconducting phase (SC) in the centre.
bring in an extra electron as shown in Figure 25? It will go in one of the empty pockets, and then we have only p�1 pockets left to scatter to reducing the binding energy to�4ℏd2Np p�1ωdð Þwith a changeΔ¼4ℏdω2dNp. Therefore, the new electron raises the energy byΔ, and therefore to offset this increase of energy, we have to apply a voltage as big asΔfor tunneling to happen.
8. HighTcin cuprates
High-temperature superconductors (abbreviated high-Tcor HTS) are materials that behave as superconductors at unusually high temperatures. The first high-Tc
superconductor was discovered in 1986 by IBM researchers Georg Bednorz and K.
Alex Müller, who were awarded the 1987 Nobel Prize in Physics “for their impor- tant break-through in the discovery of superconductivity in ceramic materials”.
Whereas “ordinary” or metallic superconductors usually have transition temperatures (temperatures below which they are superconductive) below 30 K (243.2°C) and must be cooled using liquid helium in order to achieve
superconductivity, HTS have been observed with transition temperatures as high as 138 K (135°C) and can be cooled to superconductivity using liquid nitrogen.
Compounds of copper and oxygen (so-called cuprates) are known to have HTS properties, and the term high-temperature superconductor was used interchange- ably with cuprate superconductor. Examples are compounds such as lanthanum strontium copper oxide (LASCO) and neodymium cerium copper oxide (NSCO).
Figure 24.
(A) Depiction on how a tiny voltage between two metals separated by an insulating barrier generates current that goes through an insulating barrier through tunneling. (B) If one of the metals is a superconductor, then the applied voltage has to be at least as big as the superconducting gap.
Figure 25.
Depiction on how upon tunneling an extra electron, shown in dashed lines, enters the superconducting state.
Let us take lanthanum copper oxide La2CuO4, where lanthanum donates three electrons and copper donates two and oxygen accepts two electrons and all valence is satisfied. The Cu is in state Cu2+with electrons in d9configuration [11]. d orbitals are all degenerated, but due to crystal field splitting, this degeneracy is broken, and dx2y2 orbital has the highest energy and gets only one electron (the ninth one) and forms a band of its own. This band is narrow, and hence all k states get filled by only one electron each and form Wannier packets that localize electrons on their respective sites. This way electron repulsion is minimized, and in the limit U≫t (repulsion term in much larger than hopping), we have Mott insulator and antiferromagnetic phase.
When we hole/electron dope, we remove/add electron to dx2y2band. For example, La2xSrxCuO4is hole doped as Sr has valence 2 and its presence further removes electrons from copper. Nd2xCexCuO4is electron doped as Ce has valence 4 and its presence adds electrons from copper. These extra holes/electrons form packets. When we discussed superconductivity, we discussed packets of widthωd. In d-bands, a packet of this width has many more k states as d-bands are narrow and only 1–2 eV thick. This means N (k-points in a packet) is very large and we have much larger gapΔand Tc. This is a way to understand high Tc, d-wave packets with huge bandwidths. This is as shown in Figure 26. As we increase doping and add
Figure 26.
(A) Depiction on how in narrowdband the electron wave packet comprises allkpoints to minimize repulsion.
(B) The wave packet in two dimensions with wave packet formed from superposition of k-points (along the k- direction) inside the Fermi sphere.
Figure 27.
The characteristic phase diagram of a highTccuprate. Shown are the antiferromagnetic insulator (AF) phase on the left anddome-shaped superconducting phase (SC) in the centre.
more electrons, the packet width further increases till it is≫ωd; then, we have significant offset evolution, and binding is hurt leading to loss of superconductivity.
This explains the dome characteristic of superconducting phase, whereby supercon- ductivity increases and then decreases with doping. The superconducting dome is shown in Figure 27.
Author details Navin Khaneja
IIT Bombay, Powai, India
*Address all correspondence to: [email protected]
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/
by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]Tinkham M. Introduction to Superconductivity. 2nd ed. New York, USA: McGraw Hill; 1996
[2]De Gennes PG. Superconductivity of Metals and Alloys. New York, USA:
W.A. Benjamin, Inc; 1966
[3]Bardeen J, Cooper L, Schriffer JR.
Theory of superconductivity. Physical Review. 1957;108(5):1175
[4]Kittel C. Introduction to Solid State Physics. 8th ed. John Wiley and Sons;
2005
[5]Ashcroft NW, Mermin D. On a new method in the theory of
superconductivity. Solid State Physics.
Harcourt College Publishers; 1976 [6]Simon S. Oxford Solid State Basics.
New Delhi, India: Oxford University Press; 2013
[7]Kittel C, Kroemer H. Thermal Physics. Orlando, USA: Freeman and Co; 2002
[8]Bogoliubov NN. Nuovo Cimento.
1958;7:794
[9]Deaver B, Fairbank W. Experimental evidence for quantized flux in
superconducting cylinders. Physical Review Letters. 1961;7(2):43-46 [10]Doll R, Näbauer M. Experimental proof of magnetic flux quantization in a superconducting ring. Physical Review Letters. 1961;7(2):51-52
[11]Khomskii D. Transition Metal Compounds. Cambridge University Press; 2014
more electrons, the packet width further increases till it is≫ωd; then, we have significant offset evolution, and binding is hurt leading to loss of superconductivity.
This explains the dome characteristic of superconducting phase, whereby supercon- ductivity increases and then decreases with doping. The superconducting dome is shown in Figure 27.
Author details Navin Khaneja
IIT Bombay, Powai, India
*Address all correspondence to: [email protected]
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/
by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]Tinkham M. Introduction to Superconductivity. 2nd ed. New York, USA: McGraw Hill; 1996
[2]De Gennes PG. Superconductivity of Metals and Alloys. New York, USA:
W.A. Benjamin, Inc; 1966
[3]Bardeen J, Cooper L, Schriffer JR.
Theory of superconductivity. Physical Review. 1957;108(5):1175
[4]Kittel C. Introduction to Solid State Physics. 8th ed. John Wiley and Sons;
2005
[5]Ashcroft NW, Mermin D. On a new method in the theory of
superconductivity. Solid State Physics.
Harcourt College Publishers; 1976 [6]Simon S. Oxford Solid State Basics.
New Delhi, India: Oxford University Press; 2013
[7]Kittel C, Kroemer H. Thermal Physics. Orlando, USA: Freeman and Co; 2002
[8]Bogoliubov NN. Nuovo Cimento.
1958;7:794
[9]Deaver B, Fairbank W. Experimental evidence for quantized flux in
superconducting cylinders. Physical Review Letters. 1961;7(2):43-46 [10]Doll R, Näbauer M. Experimental proof of magnetic flux quantization in a superconducting ring. Physical Review Letters. 1961;7(2):51-52
[11]Khomskii D. Transition Metal Compounds. Cambridge University Press; 2014